An update: I checked my mail today, and there was a plain brown box with $1500 cash in it, sitting in front of my door! No signature was required. That's a bit scary, but no one stole the money, so it's OK.
The box was shipped UPS Ground. I checked my receipt, and I actually wasn't charged for expedited shipping. I could've sworn I selected expedited shipping -- maybe they don't actually offer it for this program, even though there is a place to select it?
Oddly enough, although I have the cash in hand (well, not literally
in hand -- I couldn't fit that many coins in my hands), my credit card has still not been charged. I hope they charge it in time to qualify for the 15000 Starpoints promotion, rather than, say, the next day!